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Water Wave

Theme C - Wave Behaviour 

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Rocket

Theme A - Space, Time & Motion

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Thermometer

Theme B - The Particulate Nature Of Matter

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Magnifying Glass Tilted Right

Theme D - Fields

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Microscope

Theme E - Nuclear & Quantum Physics

Let&#39;s swing into the world of pendulums and oscillations!

<ul>
	<li>What is it? An object on the end of a string, swinging in a vertical plane.</li>
	<li>Harmonic Motion? Yep, but the swing angle from vertical should be &lt;10&deg;.</li>
	<li>String Length: l</li>
	<li>Mass of Bob: m</li>
	<li>Swing Angle: &theta;</li>
</ul>

<ul>
	<li>The force that pulls the pendulum bob back is &minus;mg sin (&theta;).
	<ul>
		<li>Why negative? Angle measured anticlockwise, but force acts clockwise!</li>
	</ul>
	</li>
</ul>

🔍 Analysis

<ul>
	<li>&minus;mgsin(&theta;) = ma leading to a = &minus; g sin (&theta;)</li>
	<li>Length of the arc from equilibrium to bob is x. Thus, &theta; =\(\frac xl\)​</li>
	<li>So,&nbsp; a = &minus; \(\frac gl\)​x
	<ul>
		<li>Fun fact: If &theta; &lt; 10&deg;, sin (&theta;) and &theta; are close (within 1%)!</li>
	</ul>
	</li>
</ul>

<ul>
	<li>T=&nbsp;&nbsp;\(\frac { 2π \sqrt l} g\)</li>
</ul>

🌊 Similar Analyses: Oscillators like floating cylinders on lakes!

<ul>
	<li>What if amplitude &gt;10&deg;? (Simple: The time period T deviates!)</li>
	<li>Why not measure T of a 10&deg; pendulum to 3 decimal places? (Hint: Preciseness &ne; Accuracy!)</li>
	<li>Designing an experiment? Timing over several oscillations helps reduce errors from reaction time.</li>
</ul>

🌏 Real-world Application: Grandfather clocks use pendulums to maintain accurate time, but they work best when the swings aren&#39;t too wild!

<ul>
	<li>Object&#39;s Acceleration vs. Displacement
	<ul>
		<li>Simple Harmonic Motion: If acceleration is proportional to negative displacement.</li>
		<li>Amplitude: Maximum displacement (e.g., 5.0cm).</li>
		<li>Period: T = \(\frac {2π }{w}\)​ where &omega; is related to graph&#39;s slope.</li>
	</ul>
	</li>
	<li>Mass-Spring System
	<ul>
		<li>Frequency: \(f = \frac {1}t\) where \(T = ​2π​​ \sqrt \frac mk\)</li>
		<li>Max Force: Fmax ​= m&omega;2x0​ where x0​ is amplitude.</li>
	</ul>
	</li>
	<li>Another Mass-Spring System
	<ul>
		<li>Can you use the given details to find angular frequency, max force, and spring constant? Try it out!</li>
	</ul>
	</li>
</ul>

<ul>
	<li>When is the motion simple harmonic based on force and displacement relations?</li>
	<li>Can you compute the period of a pendulum with a given length? Or vice versa?</li>
	<li>Given mass and spring constant, determine the period and max force for a spring system.</li>
	<li>How would you find the spring constant of a mass-spring system if given its frequency?</li>
	<li>A vertically hanging spring stretches by a distance when a mass is attached. Can you find this distance and the force when it&#39;s displaced?</li>
</ul>

🖼 Real-world Application: Think of a trampoline! When you jump, the force you exert stretches it, and it restores you to your original position. A playful example of oscillations!

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Happy Studying! 🎉📖🚀

Theme C - Wave Behaviour

Simple Pendulum & Its Harmonic Motion Explained

Table of content

Simple pendulum 🕰

Restoring force 📈

Period of swing 🧮

Unlock the Full Content!

Theme C - Wave Behaviour

Simple Pendulum & Its Harmonic Motion Explained

Table of content

Simple pendulum 🕰

Restoring force 📈

Period of swing 🧮

Unlock the Full Content!